After completing the main construction phase of Wendelstein 7-X (W7-X) and successfully commissioning the device, first plasma operation started at the end of 2015. Integral commissioning of plasma start-up and operation using electron cyclotron resonance heating (ECRH) and an extensive set of plasma diagnostics have been completed, allowing initial physics studies during the first operational campaign. Both in helium and hydrogen, plasma breakdown was easily achieved. Gaining experience with plasma vessel conditioning, discharge lengths could be extended gradually. Eventually, discharges lasted up to 6 s, reaching an injected energy of 4 MJ, which is twice the limit originally agreed for the limiter configuration employed during the first operational campaign. At power levels of 4 MW central electron densities reached 3  ×  1019 m−3, central electron temperatures reached values of 7 keV and ion temperatures reached just above 2 keV. Important physics studies during this first operational phase include a first assessment of power balance and energy confinement, ECRH power deposition experiments, 2nd harmonic O-mode ECRH using multi-pass absorption, and current drive experiments using electron cyclotron current drive. As in many plasma discharges the electron temperature exceeds the ion temperature significantly, these plasmas are governed by core electron root confinement showing a strong positive electric field in the plasma centre.

We investigate the existence of powers of Hamiltonian cycles in graphs with large minimum degree to which some additional edges have been added in a random manner. For all integers k ⩾ 1 , r ⩾ 0, and ℓ ⩾ ( r + 1 ) r, and for any α > k k + 1 we show that adding O ( n 2 − 2 ∕ ℓ ) random edges to an n‐vertex graph G with minimum degree at least α n yields, with probability close to one, the existence of the ( k ℓ + r )‐th power of a Hamiltonian cycle. In particular, for r = 1 and ℓ = 2 this implies that adding O ( n ) random edges to such a graph G already ensures the ( 2 k + 1 )‐st power of a Hamiltonian cycle (proved independently by Nenadov and Trujić). In this instance and for several other choices of k , ℓ, and r we can show that our result is asymptotically optimal.

The size-Ramsey numberR(F ) of a graph F is the smallest integer m such that there exists a graph G on m edges with the property that any colouring of the edges of G with two colours yields a monochromatic copy of F . In this paper, first we focus on the size-Ramsey number of a path P n on n vertices. In particular, we show that 5n/2 − 15/2 ≤R(P n ) ≤ 74n for n sufficiently large. (The upper bound uses expansion properties of random d-regular graphs.) This improves the previous lower bound,R(P n ) ≥ (1 + √ 2)n − O(1), due to Bollobás, and the upper bound,R(P n ) ≤ 91n, due to Letzter. Next we study long monochromatic paths in edge-coloured random graph G(n, p) with pn → ∞. Let α > 0 be an arbitrarily small constant. Recently, Letzter showed that a.a.s. any 2-edge colouring of G(n, p) yields a monochromatic path of length (2/3 − α)n, which is optimal. Extending this result, we show that a.a.s. any 3-edge colouring of G(n, p) yields a monochromatic path of length (1/2 − α)n, which is also optimal. In general, we prove that for r ≥ 4 a.a.s. any r-edge colouring of G(n, p) yields a monochromatic path of length (1/r − α)n. We also consider a related problem and show that for any r ≥ 2, a.a.s. any r-edge colouring of G(n, p) yields a monochromatic connected subgraph on (1/(r−1)−α)n vertices, which is also tight.

The next step in the Wendelstein stellarator line is the large superconducting device Wendelstein 7-X, currently under construction in Greifswald, Germany. Steady-state operation is an intrinsic feature of stellarators, and one key element of the Wendelstein 7-X mission is to demonstrate steady-state operation under plasma conditions relevant for a fusion power plant. Steady-state operation of a fusion device, on the one hand, requires the implementation of special technologies, giving rise to technical challenges during the design, fabrication and assembly of such a device. On the other hand, also the physics development of steady-state operation at high plasma performance poses a challenge and careful preparation. The electron cyclotron resonance heating system, diagnostics, experiment control and data acquisition are prepared for plasma operation lasting 30 min. This requires many new technological approaches for plasma heating and diagnostics as well as new concepts for experiment control and data acquisition.

In the random k-uniform hypergraph H n,p;k of order n each possible k-tuple appears independently with probability p. A loose Hamilton cycle is a cycle of order n in which every pair of adjacent edges intersects in a single vertex. We prove that if pn k−1 / log n tends to infinity with n then lim n→∞ 2(k−1)|n Pr(H n,p;k contains a loose Hamilton cycle) = 1. This is asymptotically best possible.